Blow-up of positive solutions for a family of nonlinear parabolic equations in general domain in RN

A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton–Jacobi equations

“…This is called gradient blow-up (GBU) and it is also known that T < ∞ whenever u 0 is suitably large, whereas solutions exist globally and decay to 0 if u 0 C 1 is sufficiently small (see e.g. [3], [1], [41], [23]). The singular set, or GBU set, of u is defined by GBU S(u 0 ) = x 0 ∈ Ω; lim sup t→T − , x→x 0 |∇u(x, t)| = ∞ and the elements of GBU S(u 0 ) are called GBU points.…”

Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

“…The main equation they studied is the heat equation of the form u t − ∆u = |u| p−1 u in bounded or unbounded smooth domain in R N . The theory of blowup for heat equation is already developed, we refer the reader to [16,19,25,26,28] and the references therein. While for the p-Laplacian equations of the form u t − ∆ p u = f (x, t, u, ∇u), there are still many problems worth studying, such as the blowup rate, the blowup time estimate, the asymptotic behavior of blowup solutions, the blowup criteria and so on.…”

A semilinear equation with generalized Wentzell boundary condition

“…The finite time blow-up conditions for the solutions for the equivalent Cauchy problem were first considered by Fujita in [1] for the case a(x) ≡ 1. Since then, the finite time blow-up of solutions has been extensively investigated for the Dirichlet and Neumann boundary conditions (see [2][3][4][5][6], and the references therein). The motivation to study this equation with the generalized Wentzell boundary conditions stems from the understanding that from the probabilistic point of view, the Wentzell boundary conditions are the most general admissible boundary conditions of diffusion equations, since they include Dirichlet, Neumann, Robin and mixed boundary conditions as special cases [7][8][9][10][11].…”

“…The equations governing the work of Diaz and Tello follow the model proposed by Watts and Morantine [15] which consists of a parabolic type equation in a global ocean with a dynamic and diffusive nonlinear boundary condition obtained through a global energy balance for the atmospheric surface temperature. Applying the standard technique for estimating the blow-up time used in [2,1,[3][4][5][6], we have established conditions under which the solution u(x, t) must blow up in a finite time regardless of how small u 0 (x) is for the case n > 1. We have compared the estimated blow-up time with the Neumann boundary condition and established the effect of the Wentzell boundary condition over the Neumann boundary conditions and those studied in [2,1,[3][4][5][6].…”